Clustering algorithms Part 2 b Cutbased divisive clustering
Clustering algorithms: Part 2 b Cut-based & divisive clustering Pasi Fränti 17. 3. 2014 Speech & Image Processing Unit School of Computing University of Eastern Finland Joensuu, FINLAND
Part I: Cut-based clustering
Cut-based clustering • • What is cut? Can we used graph theory in clustering? Is normalized-cut useful? Are cut-based algorithms efficient?
Clustering method • Clustering method = defines the problem • Clustering algorithm = solves the problem • Problem defined as cost function – Goodness of one cluster – Similarity vs. distance – Global vs. local cost function (what is “cut”) • Solution: algorithm to solve the problem
Cut-based clustering • Usually assumes graph • Based on concept of cut • Includes implicit assumptions which are often: – No difference than clustering in vector space – Implies sub-optimal heuristics – Sometimes even false assumptions!
Cut-based clustering methods • Minimum-spanning tree based clustering (single link) • Split-and-merge (Lin&Chen TKDE 2005): Split the data set using Kmeans, then merge similar clusters based on Gaussian distribution cluster similarity. • Split-and-merge (Li, Jiu, Cot, PR 2009): Splits data into a large number of subclusters, then remove and add prototypes until no change. • DIVFRP (Zhong et al, PRL 2008): Dividing according to furthest point heuristic. • Normalized-cut (Shi&Malik, PAMI-2000): Cut-based, minimizing the disassociation between the groups and maximizing the association within the groups. • Ratio-Cut (Hagen&Kahng, 1992) • Mcut (Ding et al, ICDM 2001) • Max k-cut (Frieze&Jerrum 1997) e b to … s l i r a Det ed late add • Feng et al, PRL 2010. Particle Swarm Optimization for selecting the hyperplane.
Clustering a graph But where we get this…?
Distance graph 3 2 5 3 6 5 7 7 7 3 4 4 2 Calculate from vector space!
Space complexity of graph Distance graph 3 2 5 3 6 5 7 7 7 3 But… Complete graph 4 4 2 N∙(N-1)/2 edges = O(N 2)
Minimum spanning tree (MST) MST Distance graph 3 2 5 3 6 5 7 2 7 7 3 4 4 2 5 4 3 3 2 Works with simple examples like this
Cut Graph cut Cost function is to maximize the weight of edges cut Resulted clusters This equals to minimizing the within cluster edge weights
Cut Graph cut Resulted clusters Equivalent to minimizing MSE!
Stopping criterion Ends up to a local minimum e Di v i t vis a r ive e m o l g Ag
Clustering method
Conclusions of “Cut” • Cut Same as partition • Cut-based method Empty concept • Cut-based algorithm Same as divisive • Graph-based clustering Flawed concept • Clustering of graph more relevant topic
Part II: Divisive algorithms
Divisive approach Motivation • Efficiency of divide-and-conquer approach • Hierarchy of clusters as a result • Useful when solving the number of clusters Challenges • Design problem 1: What cluster to split? • Design problem 2: How to split? • Sub-optimal local optimization at best
Split-based (divisive) clustering
Select cluster to be split • Heuristic choices: – Cluster with highest variance (MSE) – Cluster with most skew distribution (3 rd moment) • Optimal choice: – Tentatively split all clusters Use this ! – Select the one that decreases MSE most! • Complexity of choice: – Heuristics take the time to compute the measure – Optimal choice takes only twice (2 ) more time!!! – The measures can be stored, and only two new clusters appear at each step to be calculated.
Selection example Biggest MSE… 11. 6 6. 5 7. 5 4. 3 8. 2 11. 2 … but dividing this decreases MSE more
Selection example 11. 6 6. 5 7. 5 4. 3 6. 3 8. 2 4. 1 Only two new values need to be calculated
How to split • Centroid methods: – Heuristic 1: Replace C by C- and C+ – Heuristic 2: Two furthest vectors. – Heuristic 3: Two random vectors. • Partition according to principal axis: – Calculate principal axis – Select dividing point along the axis – Divide by a hyperplane – Calculate centroids of the two sub-clusters
Splitting along principal axis pseudo code Step 1: Calculate the principal axis. Step 2: Select a dividing point. Step 3: Divide the points by a hyper plane. Step 4: Calculate centroids of the new clusters.
pe r ipa hy pla inc ing la vid ne Pr Di xis Example of dividing
Optimal dividing point pseudo code of Step 2. 1: Calculate projections on the principal axis. Step 2. 2: Sort vectors according to the projection. Step 2. 3: FOR each vector xi DO: - Divide using xi as dividing point. - Calculate distortion of subsets D 1 and D 2. Step 2. 4: Choose point minimizing D 1+D 2.
Finding dividing point • Calculating error for next dividing point: • Update centroids: e C b an in e don ) 1 ( O t !!! e im
Sub-optimality of the split
Example of splitting process 2 clusters Princ ipal a Div idin g hyp er p lane xis 3 clusters
Example of splitting process 4 clusters 5 clusters
Example of splitting process 6 clusters 7 clusters
Example of splitting process 8 clusters 9 clusters
Example of splitting process 10 clusters 11 clusters
Example of splitting process 12 clusters 13 clusters
Example of splitting process 14 clusters 15 clusters MSE = 1. 94
K-means refinement Result directly after split: MSE = 1. 94 Result after re-partition: MSE = 1. 39 Result after K-means: MSE = 1. 33
Time complexity Number of processed vectors, assuming that clusters are always split into two equal halves: Assuming unequal split to nmax and nmin sizes:
Time complexity Number of vectors processed: At each step, sorting the vectors is bottleneck:
Comparison of results Birch 1
Conclusions • Divisive algorithms are efficient • Good quality clustering • Several non-trivial design choices • Selection of dividing axis can be improved!
References 1. P Fränti, T Kaukoranta and O Nevalainen, "On the splitting method for vector quantization codebook generation", Optical Engineering, 36 (11), 3043 -3051, November 1997. 2. C-R Lin and M-S Chen, “ Combining partitional and hierarchical algorithms for robust and efficient data clustering with cohesion self-merging”, TKDE, 17(2), 2005. 3. M Liu, X Jiang, AC Kot, “A multi-prototype clustering algorithm”, Pattern Recognition, 42(2009) 689 -698. 4. J Shi and J Malik, “Normalized cuts and image segmentation”, TPAMI, 22(8), 2000. 5. L Feng, M-H Qiu, Y-X Wang, Q-L Xiang, Y-F Yang, K Liu, ”A fast divisive clustering algorithm using an improved discrete particle swarm optimizer”, Pattern Recognition Letters, 2010. 6. C Zhong, D Miao, R Wang, X Zhou, “DIVFRP: An automatic divisive hierarchical clustering method based on the furthest reference points”, Pattern Recognition Letters, 29 (2008) 2067– 2077.
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